---
title: The Harmonic Series: Why Octaves, Fifths and Thirds Come First
dek: A vibrating string or air column sounds a fundamental and a set of partials at whole-number multiples of it. Here are the first sixteen, their cents, and where they differ from equal temperament.
topic: acoustics
tags: [harmonic series, partials, acoustics]
date: 2026-10-11
updated: 2026-10-11
author: Yunus Emre Vurgun
level: intro
related: [timbre-and-overtones, just-intonation, pythagorean-tuning, equal-temperament]
faq:
  - q: What is the harmonic series?
    a: The harmonic series is the set of frequencies that are whole-number multiples of a fundamental: f, 2f, 3f, 4f and so on. A string, pipe or air column that sounds a note produces these partials along with the fundamental.
  - q: Why do the octave and fifth come before other intervals in the harmonic series?
    a: The second partial is 2f, an octave above, and the third is 3f, an octave plus a fifth. The ratios between the first few partials are the simplest ones, so the octave (2:1), fifth (3:2), fourth (4:3) and major third (5:4) appear early.
  - q: Do piano strings follow the harmonic series exactly?
    a: No. Stiff strings have upper partials that are slightly sharp of whole-number multiples, and piano tuners stretch the octaves to follow them. The series is an ideal that real instruments approximate.
  - q: How far is the seventh harmonic from the equal-tempered minor seventh?
    a: The seventh partial sounds about 968.8 cents above its octave, which is about 31 cents flat of the equal-tempered minor seventh at 1000 cents.
---

A string or air column that sounds a note produces a fundamental, and it also vibrates in parts. Those parts sound partials at whole-number multiples of the fundamental frequency: f, 2f, 3f, 4f and so on. The collection is the harmonic series. Its first intervals are the octave, the fifth, the fourth and the major third, and they are the simple ratios on which just intonation is built.

## Partials are whole-number multiples

A string held at both ends can vibrate as one loop, or as two halves, three thirds, four quarters, and so on. Each pattern has a wavelength that is a simple fraction of the first, and frequency is inversely proportional to wavelength, so each pattern sounds a whole-number multiple of the fundamental. A pipe or air column does the same, with its own end conditions determining which multiples appear. A pipe open at both ends supports the full series, while a pipe closed at one end supports only the odd multiples, which is the difference the [timbre article](/articles/timbre-and-overtones/) turns on. Those are the partials. The fundamental is the lowest, and the upper ones are the overtones in the sense used in the [timbre article](/articles/timbre-and-overtones/).

With a fundamental of 110 Hz, the first six partials are 110, 220, 330, 440, 550 and 660 Hz. The second partial doubles the frequency, so it is an octave above. The third triples it, giving an octave plus a fifth. The fourth is two octaves above the fundamental. Each partial's interval above the one before it is a simple ratio: 3:2 from partial 2 to partial 3, 4:3 from partial 3 to partial 4, 5:4 from partial 4 to partial 5.

## The first sixteen partials

The table gives each partial's position in cents above the fundamental, the nearest equal-tempered interval and the difference. A positive difference means the partial sits sharp of that interval.

| Partial | Ratio | Cents above fundamental | Nearest equal-tempered interval | Difference (cents) |
| --- | --- | --- | --- | --- |
| 1 | 1:1 | 0.0 | unison | 0.0 |
| 2 | 2:1 | 1200.0 | octave | 0.0 |
| 3 | 3:1 | 1902.0 | octave and fifth | +2.0 |
| 4 | 4:1 | 2400.0 | two octaves | 0.0 |
| 5 | 5:1 | 2786.3 | two octaves and major third | −13.7 |
| 6 | 6:1 | 3102.0 | two octaves and fifth | +2.0 |
| 7 | 7:1 | 3368.8 | two octaves and minor seventh | −31.2 |
| 8 | 8:1 | 3600.0 | three octaves | 0.0 |
| 9 | 9:1 | 3803.9 | three octaves and major second | +3.9 |
| 10 | 10:1 | 3986.3 | three octaves and major third | −13.7 |
| 11 | 11:1 | 4151.3 | between F and F♯ | +51.3 from F, −48.7 from F♯ |
| 12 | 12:1 | 4302.0 | three octaves and fifth | +2.0 |
| 13 | 13:1 | 4440.5 | three octaves and minor sixth | +40.5 |
| 14 | 14:1 | 4568.8 | three octaves and minor seventh | −31.2 |
| 15 | 15:1 | 4688.3 | three octaves and major seventh | −11.7 |
| 16 | 16:1 | 4800.0 | four octaves | 0.0 |

Read the table in three groups. Partials 2, 4 and 8 are octaves and carry no tuning error. Partials 3, 6 and 12 are fifths, each about 2 cents sharp of the equal fifth, which is the same 1.96-cent difference that separates the just and equal fifths. Partial 5 gives a major third about 14 cents flat of the equal-tempered 400, the difference that matters most in practice. Partial 7 is the first large departure from the equal scale, and partial 11 lands between F and F♯, nearly halfway. The spacing also shrinks as the series rises. The gap from partial 8 to partial 9 is about 204 cents, and the gap from partial 15 to partial 16 is about 112 cents, so the upper partials crowd together.

The intervals also run in a fixed order. Partial 3 to partial 4 is a perfect fourth, which is the step from G3 up to C4. Played in equal temperament, it is the same 500-cent interval that the just 4:3 gives within 2 cents:

[keys]G3 C4[/keys]

## The series on a single note

The series built on C2 gives the first eight partials as pitches. Each one is written at its nearest equal-tempered pitch and played with its offset in cents: partials 3 and 6 about 2 cents sharp, partial 5 about 14 cents flat, and partial 7 about 31 cents flat, so the seventh is B♭4 lowered by 31 cents. Played as a sequence, the partials sound like a melody; in a real instrument they fuse into one tone:

[keys bpm=80 voice=sine label="Partials 1 to 8 of C2"]C2 C3 G3~2 C4 E4~-14 G4~2 Bb4~-31 C5[/keys]

## Why simple ratios come early

Intervals made from small whole numbers have partials that coincide often. Take the octave, 2:1: the second partial of the lower note is the first partial of the higher one. The fifth, 3:2, shares every third partial with the lower note and every second partial with the higher. A common explanation of consonance rests on this coincidence, and the harmonic series is where the ratios come from. The series does not by itself say which intervals sound good; it says why small ratios are the natural ones.

The partials also fuse. The ear hears a stack of partials as one note, usually at the fundamental's pitch, even when the fundamental itself is weak or missing. That fusion is why a sustained chord of partials, like the one above, can be heard as a single sound rather than a cluster of separate pitches.

Stacked, the first six partials of C3 form a single sound. The fifth partial is E5 played 14 cents flat, and the third and sixth partials are G4 and G5 played 2 cents sharp:

[keys voice=sine label="First six partials of C3 stacked"]C3+C4+G4~2+C5+E5~-14+G5~2[/keys]

## Natural horns, trumpets and bugles

Natural horns, natural trumpets and bugles have no valves, so their notes are partials. The player changes lip tension to select which partial sounds, and the instrument's tube supplies the fundamental. The calls and fanfares written for these instruments are built from the partials the player can reach. The first, third and fifth partials of C are the notes of a C major triad, which is one reason the music of these instruments leans on that chord.

The same series explains the limits of the instrument. A natural horn can reach the seventh partial, but that partial sits about 31 cents below the equal-tempered minor seventh, so players adjust it with the lips to make it usable. The higher partials are closer together, which makes the upper register crowded and demanding. The example below uses partials 3, 4, 5, 6 and 8 of the series on C3, the low fundamental of a natural trumpet in C, as a short call:

[keys]G4 C5 E5 G5 C6[/keys]

## Where the series misleads people

The series is an ideal. A real string is stiff, and its upper partials drift sharp of exact multiples. Piano tuners stretch the octaves to follow those partials, so the equal-tempered piano is not a pure harmonic sound. Bells are far less harmonic than strings, and their partials are spread much further from whole-number multiples.

The series is also not a scale. Tuning systems choose which partials and which ratios to use. Pythagorean tuning uses the chain of 3:2 fifths, which comes from partials 2 and 3 and their folding into an octave. Just intonation uses thirds such as 5:4 and 6:5, which come from partials 4, 5 and 6. The series supplies the ratios, and each system decides which ones to keep.

Finally, the eleventh partial and the partials beyond it are not the same as the just intervals of common-practice harmony. Most Western tonal music ignores the eleventh, which lies between F and F♯, and the thirteenth, which sits between A♭ and A.

## What to do with it

The string harmonics of a guitar make the series audible. Rest a finger lightly on the low E string over the 12th fret, without pressing it to the fret, and pluck the string with the other hand. A light touch at the 12th fret sounds the octave. Touching at the 7th fret sounds the octave and a fifth. Touching at the 5th fret sounds two octaves. The 7th-fret harmonic should read close to B3, and a tuner will show it within a few cents of the equal-tempered B.

The same method works on any string instrument with a clear fundamental. Play the harmonic at the midpoint, then at a third and a quarter of the string's length, and listen for the intervals without naming them first. Name them only after you have heard them.

To check the reading, use a tuner with the open string as the reference. The harmonic at the 7th fret should sit a few cents above the equal-tempered B, and the 5th-fret harmonic should sit close to the equal E4. If one of them is well off, the usual cause is that the finger is touching slightly away from the node, so move it until the tone is clearest and the tuner settles.